The semiclassical limit from the Pauli-Poisswell/Darwin to the Euler-Poisswell/Darwin system by WKB methods

TL;DR

This paper derives the semiclassical limit of semi-relativistic Pauli equations to classical Euler-Poisswell/Darwin systems using WKB methods, connecting quantum spin models to classical plasma dynamics.

Contribution

It establishes the rigorous semiclassical limit from Pauli-Poisswell/Darwin equations to Euler-Poisswell/Darwin systems, including convergence results and well-posedness analysis.

Findings

Proved local in time semiclassical limit $ abla ightarrow 0$ to Euler-Poisswell/Darwin equations.

Established weak convergence of Wigner transforms to Vlasov-type equations.

Proved local well-posedness and blow-up criteria for the Euler-Poisswell/Darwin equations.

Abstract

The self-consistent Pauli-Poisswell and Pauli-Darwin equations for 2-spinors are $O(1/c)$ (where $c$ denotes the speed of light) semi-relativistic approximations of the Dirac-Maxwell equation for 4-spinors coupled to the self-consistent electromagnetic fields generated by the charge and current densities of a fast moving electric charge. They consist of a vector-valued magnetic Schr\"odinger equation with the Stern-Gerlach term which couples spin and magnetic field, coupled to 1+3 Poisson equations as the magnetostatic approximation of Maxwell's equations. The Pauli-Poisswell and Pauli-Dariwn euqations are $O(1/c)$ models keeping both relativistic effects magnetism and spin, both of which are absent in the non-relativistic Schr\"odinger-Poisson equation and inconsistent in the magnetic Schr\"odinger-Maxwell equation. We prove the local in time semiclassical limit $\hbar \rightarrow 0$ to the Euler-Poisswell equation and Euler-Darwin equations based on WKB analysis and energy estimates. Moreover we obtain weak convergence of the monokinetic Wigner transform to the monokinetic scalar Wigner measure solving the Vlasov-Poisswell and Vlasov-Darwin equations and strong convergence of the macroscopic densities. We introduce the Euler-Poisswell/Darwin equation and prove local wellposedness and a blow up alternative.